Grassmannians and Other Schubert Varieties

Sara Billey University of Washington http://www.math.washington.edu/ billey ∼ What is Algebraic Geometry? April 14, 2007 Famous Quotations

Arnold Ross (and PROMYS). ”Think deeply of simple things.”

Angela Gibney. Why do algebraic geometers love moduli spaces? “It is just like with people, if you want to get to know someone, go to their family reunion.”

Goal. Focus our attention on a particular family of varieties which are indexed by combinatorial data where lots is known about their structure and yet lots is still open. Schubert Varieties

A Schubert variety is a member of a family of projective varieties whose points are indexed by matrices and whose deﬁning equations are determinantal minors.

Typical properties: • This family of varieties is indexed by combinatorial objects; e.g. partitions, permutations, or Weyl group elements. • Some are smooth and some are singular. • Their cohomology rings include familiar subsets and quotients of polynomial rings including the symmetric functions. Enumerative Geometry

Approximately 150 years ago. . . Grassmann, Schubert, Pieri, Giambelli, Severi, and others began the study of enumerative geometry.

Early questions: • What is the dimension of the intersection between two general lines in R2? • How many lines intersect two given lines and a given point in R3?

• How many lines intersect four given lines in R3 ?

Modern questions:

• How many points are in the intersection of 2,3,4,. . . Schubert varieties in general position? Why Study Schubert Varieties?

1. It can be useful to see points, lines, planes etc as families with certain properties. 2. Schubert varieties provide interesting examples for test cases and future research in algebraic geometry. 3. Applications in discrete geometry, computer graphics, and computer vision. Outline

1. Review of vector spaces and projective spaces

2. Introduction to Grassmannians, Flag Manifolds and Schubert Varieties 3. Five Fun Facts on Schubert Varieties

4. Permutation Arrays = higher dimensional analog of permutation matrices 5. Solving Schubert problems with Permutation Arrays Vector Spaces

• V is a vector space over a ﬁeld F if it is closed under addition and multiplication by scalars in F. • B = b ,...,b is a basis for V if for every a V there exist unique { 1 k} ∈ scalars c ,...,c F such that 1 k ∈ a = c b + c b + + c b = (c , c ,...,c ) Fk. 1 1 2 2 ··· k k 1 2 k ∈ • The dimension of V equals the size of a basis. • A subspace U of a vector space V is any subset of the vectors in V that is closed under addition and scalar multiplication.

Fact. Any basis for U can be extended to a basis for V . Projective Spaces

Defn. • P(V ) = 1-dim subspaces of V = V . { } hd·a=ai

1-dim subspaces = lines in V through 0 points in P(V ) ←→ 2-dim subspaces = planes in V through 0 lines in P(V ) ←→

• Given a basis B = b ,b ,...,b for V , the line spanned by { 1 2 k} a = c b + c b + + c b P(V ) has homogeneous coordinates 1 1 2 2 ··· k k ∈ [c : c : : c ] = [dc : dc : : dc ] for any d F. 1 2 ··· k 1 2 ··· k ∈ The Grassmannian Varieties

Deﬁnition. Fix a vector space V over C (or R, Z2,. . . ) with basis B = e ,...,e . The Grassmannian variety { 1 n} G(k,n) = k-dimensional subspaces of V . { }

Question.

How can we impose the structure of a variety or a manifold on this set? The Grassmannian Varieties

Answer. Relate G(k,n) to the vector space of k n matrices. ×

U =span 6e + 3e , 4e + 2e , 9e + e + e G(3, 4) h 1 2 1 3 1 3 4i ∈

6 3 0 0 MU =  4 0 2 0  9 0 1 1  

• U G(k,n) rows of M are independent vectors in V ∈ ⇐⇒ U ⇐⇒ some k k minor of M is NOT zero. × U Pl¨ucker Coordinates • Deﬁne fj1,j2,...,jk to be the polynomial given by the determinant of the matrix

x1,j1 x1,j2 ... x1,jk

x2,j1 x2,j2 ... x2,jk  . . . .  . . . .    xkj xkj ... xkjk   1 2  • G(k,n) is an open set in the Zariski topology on k n matrices deﬁned × as the complement of the union over all k-subsets of 1, 2,...,n of { } the varieties V (fj1,j2,...,jk ). • All the determinants fj1,j2,...,jk are homogeneous polynomials of degree k so G(k,n) can be thought of as an open set in projective space. The Grassmannian Varieties

Canonical Form. Every subspace in G(k,n) can be represented by a unique matrix in row echelon form.

Example.

U =span 6e + 3e , 4e + 2e , 9e + e + e G(3, 4) h 1 2 1 3 1 3 4i ∈

6 3 0 0 3 0 0 2 1 0 0  4 0 2 0  =  0 2 0   2 0 1 0  ≈ 9 0 1 1 0 1 1 7 0 0 1      

2e + e , 2e + e , 7e + e ≈h 1 2 1 3 1 4i Subspaces and Subsets

Example.

5 9 10000000h U = RowSpan 580979 1h 0 0 0 G(3, 10). ∈ 46026403 1h 0   position(U) = 3, 7, 9 { } Deﬁnition.

If U G(k,n) and M is the corresponding matrix in canonical form then ∈ U the columns of the leading 1’s of the rows of MU determine a subset of size k in 1, 2,...,n := [n]. There are 0’s to the right of each leading 1 and 0’s { } above and below each leading 1. This k-subset determines the position of U with respect to the ﬁxed basis. The Schubert Cell Cj in G(k,n)

Defn. Let j = j < j < < jk [n]. A Schubert cell is { 1 2 ··· } ∈

C = U G(k,n) position(U) = j ,...,j j { ∈ | { 1 k}}

Fact. G(k,n) = Cj over all k-subsets of [n] [

Example. In G(3, 10),

10000000h ∗ ∗ C{3,7,9} =  0 1h 0 0 0 ∗ ∗ ∗ ∗ ∗  0 0 1h 0  ∗ ∗ ∗ ∗ ∗ ∗    • Observe dim(C{3,7,9}) = 13. • In general, dim(C ) = j i. j i − P Schubert Varieties in G(k,n)

Defn. Let j = j < j < < jk [n]. A Schubert variety is { 1 2 ··· } ∈

Xj = Closure of Cj under Zariski topology.

Question. In G(3, 10), what polynomials vanish on C{3,7,9}?

10000000h ∗ ∗ C{3,7,9} =  0 1h 0 0 0 ∗ ∗ ∗ ∗ ∗  0 0 1h 0  ∗ ∗ ∗ ∗ ∗ ∗    4 j 8 ≤ 1 ≤ Answer. All minors fj ,j ,j with  or j = 3 and 8 j 9  1 2 3 1 ≤ 2 ≤  or j1 = 3, j2 = 7 and j3 = 10    In other words, the canonical form for any subspace in Cj has 0’s to the right of column ji in each row i. k-Subsets and Partitions

Defn. A partition of a number n is a weakly increasing sequence of nonnegative integers λ = (λ λ λ ) 1 ≤ 2 ≤···≤ k such that n = λ = λ . i | | P Partitions can be visualized by their Ferrers diagram

(2, 5, 6) −→

Fact. There is a bijection between k-subsets of 1, 2,...,n and partitions { } whose Ferrers diagram is contained in the k (n k) rectangle given by × − shape : j <...

Defn. A partial order or a poset is a reﬂexive, anti-symmetric, and transitive relation on a set.

Defn. Young’s Lattice If λ = (λ λ λ ) and µ = (µ µ µ ) then 1 ≤ 2 ≤ ··· ≤ k 1 ≤ 2 ≤ ··· ≤ k λ µ if the Ferrers diagram for λ ﬁts inside the Ferrers diagram for µ. ⊂

⊂ ⊂

Facts.

1. Xj = Ci. shape(i)[⊂shape(j) 2. The dimension of X is shape(j) . j | |

3. The Grassmannian G(k,n) = X{n−k+1,...,n−1,n} is a Schubert variety! Enumerative Geometry Revisited

Question. How many lines intersect four given lines in R3 ?

Translation. Given a line in R3, the family of lines intersecting it can be interpreted in G(2, 4) as the Schubert variety

1 0 0 X , = ∗ {2 4} 0 1 ∗ ∗ with respect to a suitably chosen basis determined by the line.

Reformulated Question. How many subspaces U G(2, 4) are in ∈ the intersection of 4 copies of the Schubert variety X{2,4} each with respect to a diﬀerent basis?

Modern Solution. Use Intersection Theory! Intersection Theory

• Schubert varieties induce canonical elements of the cohomology ring for G(k,n) called Schubert classes: [Xj]. • Multiplication of Schubert classes corresponds with intersecting Schubert varieties with respect to diﬀerent bases.

[X ][X ] = X (B1) X (B2) i j i ∩ j • Schubert classes add and multiply just like Schur functions. Schur functions are a fascinating family of symmetric functions indexed by partitions which appear in many areas of math, physics, theoretical computer science. • Expanding the product of two Schur functions into the basis of Schur functions can be done via linear algebra:

ν SλSµ = cλ,µSν . X • ν The coefficients cλ,µ are non-negative integers called the Littlewood- Richardson coefficients. In a 0-dimensional intersection, the coefficient of [X ] is the number of subspaces in X (B1) X (B2). {1,2,...,k} i ∩ j Enumerative Solution

Reformulated Question. How many subspaces U G(2, 4) are in ∈ the intersection of 4 copies of the Schubert variety X{2,4} each with respect to a diﬀerent basis? Enumerative Solution

Reformulated Question. How many subspaces U G(2, 4) are in ∈ the intersection of 4 copies of the Schubert variety X{2,4} each with respect to a diﬀerent basis?

Solution. X{2,4} = S(1) = x1 + x2 + ... By the recipe, compute

X (B1) X (B2) X (B3) X (B4) {2,4} ∩ {2,4} ∩ {2,4} ∩ {2,4} 4 = S(1) = 2S(2,2) + S(3,1) + S(2,1,1).

Answer. The coeﬃcient of S2,2 = [X1,2] is 2 representing the two lines meeting 4 given lines in general position. Recap

1. G(k,n) is the Grassmannian variety of k-dim subspaces in Rn.

2. The Schubert varieties in G(k,n) are nice projective varieties indexed by k-subsets of [n] or equivalently by partitions in the k (n k) rectangle. × − 3. Geometrical information about a Schubert variety can be determined by the combinatorics of partitions.

4. Intersection theory applied to Schubert varieties can be used to solve problems in enumerative geometry. Generalizations

”Nothing is more disagreeable to the hacker than duplication of eﬀort. The ﬁrst and most important mental habit that people develop when they learn how to write computer programs is to generalize, generalize, generalize.: –Neil Stephenson ”In the Beginning was the Command Line”

Same goes for mathematicians! The Flag Manifold

n Defn. A complete ﬂag F• = (F1,...,Fn) in C is a nested sequence of vector spaces such that dim(F ) = i for 1 i n. F is determined by an i ≤ ≤ • ordered basis f ,f ,...f where F = span f ,...,f . h 1 2 ni i h 1 ii

Example.

F = 6e + 3e , 4e + 2e , 9e + e + e , e • h 1 2 1 3 1 3 4 2i

The Flag Manifold

Canonical Form.

F = 6e + 3e , 4e + 2e , 9e + e + e , e • h 1 2 1 3 1 3 4 2i

6 3 0 0 300 0 2 1 0 0 4 0 2 0 020 0 2 0 1 0   =     ≈ 9 0 1 1 011 0 7 0 0 1        0 1 0 0   1 0 0 2   1 0 0 0     −   

2e + e , 2e + e , 7e + e , e ≈h 1 2 1 3 1 4 1i

l (C) := flag manifold over Cn n G(n,k) F n ⊂ k=1 Q = complete flags F { •} = B GL (C), B = lower triangular mats. \ n Flags and Permutations 2 1h 0 0 2 0 1h 0 Example. F = 2e +e , 2e +e , 7e +e , e   • h 1 2 1 3 1 4 1i ≈ 7 0 0 1h    1h 0 0 0    Note. If a flag is written in canonical form, the positions of the leading 1’s form a permutation matrix. There are 0’s to the right and below each leading 1. This permutation determines the position of the flag F• with respect to the reference flag E = e , e , e , e . • h 1 2 3 4 i Many ways to represent a permutation

0 1 0 0 0 1 1 1 0 0 1 0 1 2 3 4 0 1 2 2   = = 2341 =   0 0 0 1 2 3 4 1 0 1 2 3      1 0 0 0   1 2 3 4      matrix two-line one-line rank notation notation notation table

... 1234 ∗ ... ∗ = = (1, 2, 3) ... ∗ .... 2341

diagram of a reduced string diagram permutation word The Schubert Cell Cw(E ) in ln(C) • F

Defn. Cw(E•) = All ﬂags F• with position(E•,F•) = w

= F l dim(E F ) = rk(w[i, j]) { • ∈ F n | i ∩ j }

2 1h 0 0 1 0 0 ∗  2 0 1h 0   0 1 0   Example. F• = C2341 = ∗ : C 7 0 0 1h ∈  0 0 1 ∗ ∈     ∗    1h 0 0 0   1 0 0 0         Easy Observations. • dimC(Cw) = l(w) = # inversions of w. • C = w B is a B-orbit using the right B action, e.g. w ·

¾ ¿ ¾ ¿ ¾ ¿ 0 1 0 0 b1,1 0 0 0 b2,1 b2,2 0 0

6 7 6 7 6 7

6 0 0 1 0 7 6 b2,1 b2,2 0 0 7 6 b3,1 b3,2 b3,3 0 7 6 7 6 7 = 6 7

6 7 6 7 6 7

6 0 0 0 1 7 6 b3,1 b3,2 b3,3 0 7 6 b4,1 b4,2 b4,3 b4,4 7

4 5 4 5 4 5

1 0 0 0 b4,1 b4,2 b4,3 b4,4 b1,1 0 0 0 The Schubert Variety Xw(E ) in ln(C) • F

Defn. Xw(E•) = Closure of Cw(E•) under the Zariski topology

= F l dim(E F ) rk(w[i, j]) { • ∈ F n | i ∩ j ≥ } where E = e , e , e , e . • h 1 2 3 4 i 1h 0 0 0 1 0 0 ∗  0 1h 0   0 1 0  Example. ∗ X2341(E•) = ∗ 0 0 1h ∈  0 0 1   ∗   ∗   0 1h 0 0   1 0 0 0        Why?. Five Fun Facts

Fact 1. The closure relation on Schubert varieties deﬁnes a nice partial order.

Xw = Cv = Xv v[≤w v[≤w Bruhat order (Ehresmann 1934, Chevalley 1958) is the transitive closure of

Example. Bruhat order on permutations in S3. 321 @ 312 @ 231 @ @ 213 @ 132 @@ 123 Observations. Self dual, rank symmetric, rank unimodal. Bruhat order on S4

4 3 2 1

3 4 2 1 4 2 3 1 4 3 1 2

2 4 3 1 3 4 1 2 3 2 4 1 4 1 3 2 4 2 1 3

1 4 3 2 2 3 4 1 2 4 1 3 3 1 4 2 3 2 1 4 4 1 2 3

1 3 4 2 1 4 2 3 2 3 1 4 2 1 4 3 3 1 2 4

1 2 4 3 1 3 2 4 2 1 3 4

1 2 3 4 Bruhat order on S5

(5 4 3 2 1)

(5 4 3 1 2) (5 4 2 3 1) (5 3 4 2 1) (4 5 3 2 1)

(5 4 2 1 3) (5 3 4 1 2) (4 5 3 1 2) (5 4 1 3 2) (5 3 2 4 1) (4 5 2 3 1) (5 2 4 3 1) (4 3 5 2 1) (3 5 4 2 1)

(5 3 2 1 4) (4 5 2 1 3) (5 4 1 2 3) (5 2 4 1 3) (4 3 5 1 2) (5 3 1 4 2) (3 5 4 1 2) (4 5 1 3 2) (5 1 4 3 2) (4 3 2 5 1) (5 2 3 4 1) (3 5 2 4 1) (4 2 5 3 1) (2 5 4 3 1) (3 4 5 2 1)

(4 3 2 1 5) (5 3 1 2 4) (5 2 3 1 4) (3 5 2 1 4) (4 5 1 2 3) (4 2 5 1 3) (5 1 4 2 3) (5 2 1 4 3) (2 5 4 1 3) (4 3 1 5 2) (3 4 5 1 2) (5 1 3 4 2) (3 5 1 4 2) (4 1 5 3 2) (1 5 4 3 2) (4 2 3 5 1) (3 4 2 5 1) (2 5 3 4 1) (3 2 5 4 1) (2 4 5 3 1)

(4 3 1 2 5) (4 2 3 1 5) (3 4 2 1 5) (5 2 1 3 4) (5 1 3 2 4) (3 5 1 2 4) (2 5 3 1 4) (3 2 5 1 4) (4 1 5 2 3) (4 2 1 5 3) (1 5 4 2 3) (5 1 2 4 3) (2 4 5 1 3) (2 5 1 4 3) (3 4 1 5 2) (4 1 3 5 2) (3 1 5 4 2) (1 5 3 4 2) (1 4 5 3 2) (3 2 4 5 1) (2 4 3 5 1) (2 3 5 4 1)

(4 2 1 3 5) (3 4 1 2 5) (4 1 3 2 5) (3 2 4 1 5) (2 4 3 1 5) (5 1 2 3 4) (2 5 1 3 4) (3 1 5 2 4) (1 5 3 2 4) (3 2 1 5 4) (2 3 5 1 4) (4 1 2 5 3) (1 4 5 2 3) (1 5 2 4 3) (2 4 1 5 3) (2 1 5 4 3) (3 1 4 5 2) (1 4 3 5 2) (1 3 5 4 2) (2 3 4 5 1)

(3 2 1 4 5) (4 1 2 3 5) (2 4 1 3 5) (3 1 4 2 5) (1 4 3 2 5) (2 3 4 1 5) (1 5 2 3 4) (2 1 5 3 4) (3 1 2 5 4) (1 3 5 2 4) (2 3 1 5 4) (1 4 2 5 3) (1 2 5 4 3) (2 1 4 5 3) (1 3 4 5 2)

(3 1 2 4 5) (2 3 1 4 5) (1 4 2 3 5) (2 1 4 3 5) (1 3 4 2 5) (1 2 5 3 4) (2 1 3 5 4) (1 3 2 5 4) (1 2 4 5 3)

(2 1 3 4 5) (1 3 2 4 5) (1 2 4 3 5) (1 2 3 5 4)

(1 2 3 4 5) 10 Fantastic Facts on Bruhat Order

1. Bruhat Order Characterizes Inclusions of Schubert Varieties

2. Contains Young’s Lattice in S∞ 3. Nicest Possible M¨obius Function

4. Beautiful Rank Generating Functions 5. [x, y] Determines the Composition Series for Verma Modules 6. Symmetric Interval [0ˆ, w] X(w) rationally smooth ⇐⇒ 7. Order Complex of (u, v) is shellable 8. Rank Symmetric, Rank Unimodal and k-Sperner 9. Eﬃcient Methods for Comparison

10. Amenable to Pattern Avoidance Singularities in Schubert Varieties

Defn. Xw is singular at a point p ⇐⇒ dimXw = l(w) < dimension of the tangent space to Xw at p.

Observation 1. Every point on a Schubert cell Cv in Xw looks locally the same. Therefore, p C is a singular point the permutation matrix v ∈ v ⇐⇒ is a singular point of Xw.

Observation 2. The singular set of a varieties is a closed set in the Zariski topology. Therefore, if v is a singular point in Xw then every point in Xv is singular. The irreducible components of the singular locus of Xw is a union of Schubert varieties:

Sing(Xw) = Xv. v∈maxsing(w)[ Singularities in Schubert Varieties

Fact 2. (Lakshmibai-Seshadri) A basis for the tangent space to Xw at v is indexed by the transpositions tij such that

vt w. ij ≤

Deﬁnitions. • Let T = invertible diagonal matrices. The T -ﬁxed points in Xw are the permutation matrices indexed by v w. ≤ • If v, vtij are permutations in Xw they are connected by a T -stable curve. The set of all T -stable curves in Xw are represented by the Bruhat graph on [id, w]. Bruhat Graph in S4

(4 3 2 1)

(4 2 3 1) (4 3 1 2) (3 4 2 1)

(4 1 3 2) (4 2 1 3) (2 4 3 1) (3 4 1 2) (3 2 4 1)

(4 1 2 3) (1 4 3 2) (2 4 1 3) (3 1 4 2) (2 3 4 1) (3 2 1 4)

(1 4 2 3) (2 1 4 3) (1 3 4 2) (3 1 2 4) (2 3 1 4)

(1 2 4 3) (2 1 3 4) (1 3 2 4)

(1 2 3 4) Tangent space of a Schubert Variety

Example. T (X ) = span xi,j tij w . 1234 4231 { | ≤ }

(4 2 3 1)

(3 2 4 1) (4 2 1 3) (4 1 3 2) (2 4 3 1)

(3 2 1 4) (3 1 4 2) (2 3 4 1) (4 1 2 3) (2 4 1 3) (1 4 3 2)

(3 1 2 4) (2 3 1 4) (2 1 4 3) (1 3 4 2) (1 4 2 3)

(2 1 3 4) (1 3 2 4) (1 2 4 3)

(1 2 3 4) dimX(4231)=5 dimT (4231) = 6 = X(4231) is singular! id ⇒ Five Fun Facts

Fact 3. There exists a simple criterion for characterizing singular Schubert varieties using pattern avoidance.

Theorem: Lakshmibai-Sandhya 1990 (see also Haiman, Ryan, Wolper) X is non-singular w has no subsequence with the same relative order w ⇐⇒ as 3412 and 4231.

w = 625431 contains 6241 4231 = X is singular ∼ ⇒ 625431 Example: w = 612543 avoids 4231 = X is non-singular ⇒ 612543 &3412 Five Fun Facts

Consequences of Fact 3.

• (Billey-Warrington, Kassel-Lascoux-Reutenauer, Manivel 2000) The bad patterns in w can also be used to eﬃciently ﬁnd the singular locus of Xw.

• (Bona 1998, Haiman) Let vn be the number of w Sn for which X(w) ∈ n is non-singular. Then the generating function V (t) = n vnt is given by P 1 5t + 3t2 + t2√1 4t V (t) = − − . 1 6t + 8t2 4t3 − − • (Billey-Postnikov 2001) Generalized pattern avoidance to all semisimple simply-connected Lie groups G and characterized smooth Schubert varieties Xw by avoiding these generalized patterns. Only requires checking patterns of types A3, B2, B3,C2,C3,D4,G2. Five Fun Facts

Fact 4. There exists a simple criterion for characterizing Gorenstein Schubert varieties using modiﬁed pattern avoidance.

Theorem: Woo-Yong (Sept. 2004)

X is Gorenstein w ⇐⇒ • w avoids 31542 and 24153 with Bruhat restrictions t ,t and { 15 23} t ,t { 15 34}

• for each descent d in w, the associated partition λd(w) has all of its inner corners on the same antidiagonal. Five Fun Facts

Fact 5. Schubert varieties are useful for studying the cohomology ring of the ﬂag manifold.

Z ∗ [x1,...,xn] Theorem (Borel): H ( ln) = . F ∼ e ,...e h 1 ni The symmetric function e = x x ...x . • i k1 k2 ki 1≤k1

[X ] w S form a basis for H∗( l ) over Z. • { w | ∈ n} F n

Question. What is the product of two basis elements?

w [Xu] [Xv] = [Xw]cuv. · X Cup Product in H ( l ) ∗ F n

Answer. Use Schubert polynomials! Due to Lascoux-Sch¨utzenberger, Bernstein- Gelfand-Gelfand, Demazure.

• BGG: If S [X ]mod e ,...e then w ≡ w h 1 ni Sw siSw − [Xwsi ] if l(w) < l(wsi) x x ≡ i − i+1 [X ] xn−1xn−2 x (x x ) ... id ≡ 1 2 ··· n−1 ≡ i − j ≡ i>jY

Here deg[Xw] = codim(Xw).

• LS: Choosing [X ] xn−1xn−2 x works best because product id ≡ 1 2 ··· n−1 expansion can be done without regard to the ideal! Schubert polynomials for S4

S w0(1234) = 1 S w0(2134) = x1 S w0(1324) = x2 + x1 S 2 w0(3124) = x1 S w0(2314) = x1x2 S 2 w0(3214) = x1x2 S w0(1243) = x3 + x2 + x1 S 2 w0(2143) = x1x3 + x1x2 + x1 S 2 2 w0(1423) = x2 + x1x2 + x1 S 3 w0(4123) = x1 S 2 2 w0(2413) = x1x2 + x1x2 S 3 w0(4213) = x1x2 S w0(1342) = x2x3 + x1x3 + x1x2 S 2 2 w0(3142) = x1x3 + x1x2 S 2 2 2 2 w0(1432) = x2x3 + x1x2x3 + x1x3 + x1x2 + x1x2 S 3 3 w0(4132) = x1x3 + x1x2 S 2 2 w0(3412) = x1x2 S 3 2 w0(4312) = x1x2 S w0(2341) = x1x2x3 S 2 w0(3241) = x1x2x3 S = x x2x + x2x x Cup Product in H ( l ) ∗ F n

Key Feature. Schubert polynomials have distinct leading terms, therefore expanding any polynomial in the basis of Schubert polynomials can be done by linear algebra just like Schur functions.

Buch: Fastest approach to multiplying Schubert polynomials uses Lascoux and Sch¨utzenberger’s transition equations. Works up to about n = 15.

w Draw Back. Schubert polynomials don’t prove cuv’s are nonnegative (ex- cept in special cases). Cup Product in H ( l ) ∗ F n Another Answer.

• By intersection theory: [X ] [X ] = [X (E ) X (F )] u · v u • ∩ v •

• Perfect pairing: [X (E )] [X (F )] [X (G )] = cw [X ] u • · v • · w0w • uv id

|| [X (E ) X (F ) X (G )] u • ∩ v • ∩ w0w • • The Schubert variety X is a single point in l . id F n

Intersection Numbers: cw = #X (E ) X (F ) X (G ) uv u • ∩ v • ∩ w0w • Assuming all ﬂags E•, F•, G• are in suﬃciently general position. Intersecting Schubert Varieties

Example. Fix three ﬂags R•, G•, and B•:

Find X (R ) X (G ) X (B ) where u, v, w are the following permu- u • ∩ v • ∩ w • tations:

R1 R2 R3 G1 G2 G3 B1 B2 B3

P 1 1 1 1 P 2 1 1 1 P 3 1 1 1 Intersecting Schubert Varieties

Example. Fix three ﬂags R•, G•, and B•:

Find X (R ) X (G ) X (B ) where u, v, w are the following permu- u • ∩ v • ∩ w • tations:

R1 R2 R3 G1 G2 G3 B1 B2 B3

P 1 1 1 1 P 2 1 1 1 P 3 1 1 1 Intersecting Schubert Varieties

Example. Fix three ﬂags R•, G•, and B•:

Find X (R ) X (G ) X (B ) where u, v, w are the following permu- u • ∩ v • ∩ w • tations:

R1 R2 R3 G1 G2 G3 B1 B2 B3

P 1 1 1 1 P 2 1 1 1 P 3 1 1 1 Intersecting Schubert Varieties

Schubert’s Problem. How many points are there usually in the intersection of d Schubert varieties if the intersection is 0-dimensional?

n • Solving approx. nd equations with variables is challenging! 2

Observation. We need more information on spans and intersections of ﬂag components, e.g. dim(E1 E2 Ed ). x1 ∩ x2 ∩···∩ xd Permutation Arrays

1 2 d Theorem. (Eriksson-Linusson, 2000) For every set of d ﬂags E• , E• ,...,E• , there exists a unique permutation array P [n]d such that ⊂ dim(E1 E2 Ed ) = rkP [x]. x1 ∩ x2 ∩···∩ xd

R1 R2 R3 R1 R2 R3 R1 R2 R3 B1 1h B2 1h 1h2 B3 1h1 1 1 1 2 1 2 3

G1 G2 G3 Totally Rankable Arrays

Defn. For P [n]d, ⊂ • rk P = # k x P s.t. x = k . j { | ∃ ∈ j } • P is rankable of rank r if rk (P ) = r for all 1 j d. j ≤ ≤ • y = (y ,...,y ) x = (x ,...,x ) if y x for each i. 1 d 1 d i ≤ i • P [x] = y P y x { ∈ | } • P is totally rankable if P [x] is rankable for all x [n]d. ∈

• X • • • 1 1 1 2 1 1 1 1 1 2 1 2 3 Union of dots is totally rankable. Including X it is not. • Permutation Arrays

O• • O• • 1 1 1 2 1 1 1 1 1 2 1 2 3

Points labeled O are redundant, i.e. including them gives another totally • rankable array with same rank table.

Defn. P [n]d is a permutation array if it is totally rankable and has no ⊂ redundant dots.

• • [4]2. ∈ • •

Open. Count the number of permutation arrays in [n]k. Permutation Arrays

Theorem. (Eriksson-Linusson) Every permutation array in [n]d+1 can be obtained from a unique permutation array in [n]d by identifying a sequence of antichains.

sh sh sh sh • sh sh • • • This produces the 3-dimensional array

P = (4, 4, 1), (2, 4, 2), (4, 2, 2), (3, 1, 3), (1, 4, 4), (2, 3, 4) . { } 4 4 2 3 2 1 Unique Permutation Array Theorem

Theorem.(Billey-Vakil 2005) If

1 d X = X 1 (E ) X d (E ) w • ∩···∩ w • is nonempty 0-dimensional intersection of d Schubert varieties with respect to 1 2 d ﬂags E• , E• ,...,E• in general position, then there exists a unique permutation array P [n]d+1 such that ∈ 1 2 d X = F• dim(E E E Fxd ) = rkP [x]. (1) { | x1 ∩ x2 ∩···∩ xd ∩ +1 } Furthermore, we can recursively solve a family of equations for X using P .

Open Problem. Can one ﬁnd a ﬁnite set of rules for moving dots in a 3-d w permutation array which determines the cuv’s analogous to one of the many Littlewood-Richardson rules?

Recent Progress. Izzet Coskun’s Mondrian tableaux. Generalizations of Schubert Calculus for G/B

1997-2007: A Highly Productive Decade.

A: GLn  B: SO2n+1  cohomology  C: SP n   quantum   2   D: SO n   equivariant   2  ×    Semisimple Lie Groups   K-theory   Kac-Moody Groups   eq. K-theory       GKM Spaces       

Contributions from: Bergeron, Billey, Brion, Buch, Carrell, Ciocan-Fontainine, Coskun, Duan, Fomin, Fulton, Gelfand, Goldin, Graham, Griﬀeth, Guillemin, Haibao, Haiman, Holm, Huber, Kirillov, Knutson, Kogan, Kostant, Kresh, S. Kumar, A. Kumar, Lam, Lascoux, Lenart, Miller, Peterson, Pitti, Postnikov, Ram, Robinson, Shimozono, Sottile, Sturmfels, Tamvakis, Vakil, Winkle, Yong, Zara... See also A. Yong’s slides on “Enumerative Formulas in Schubert Calculus” http://math.berkeley.edu/ ayong/slides.html Some Recommended Further Reading

1. “Schubert Calculus” by S. L. Kleiman; Dan Laksov The American Math- ematical Monthly, Vol. 79, No. 10. (Dec., 1972), pp. 1061-1082. 2. ”Young Tableaux” by William Fulton, London Math. Soc. Stud. Texts, Vol. 35, Cambridge Univ. Press, Cambridge, UK, 1997. 3. “The Symmetric Group” by Bruce Sagan, Wadsworth, Inc., 1991.

4. “Numerical Schubert calculus” by Birkett Huber, Frank Sottile, and Bernd Sturmfels, Journal of Symbolic Computation, 26, (1998) pp. 767-788.

5. “Determining the Lines Through Four Lines” by Michael Hohmeyer and Seth Teller, Journal of Graphics Tools, 4(3):11-22, 1999.

6. “Flag arrangements and triangulations of products of simplices” by Sara Billey and Federico Ardila, to appear in Adv. in Math. 7. Recent work of Frank Sottile and Anton Leykin here at IMA

http://www.math.tamu.edu/~sottile/pages/Flags/Data/F37/We7W2W21.91.html